to me it’s mostly about solving problems by looking at all the components of a system, and thinking about how they interact with each other to solve the problem efficiently. One important aspect of this approach is that you need to think carefully about how to assign functionality among components. The matter of whether reliable delivery should be performed at the link, network, transport, or application layer is a classic example of the sort of question that systems thinkers need to tackle.
In some sense, the Systems Approach stands in contrast to the standard “layered” model of networking, most often taught as the 7-layer OSI model. The layered model encourages you to think about one layer at a time in isolation of the others. That can be a helpful way to learn basic concepts, but layered thinking often leads to designs that may optimize one layer, and yet don’t work well in optimizing system behavior. (How useful is it to build reliable delivery into the network layer if the transport layer needs to do it all over again? Or if the application needs timely delivery more than it needs reliability?)